Hall-Littlewood polynomials, affine Schubert series, and lattice enumeration

TL;DR

This work constructs Hall--Littlewood--Schubert series $\mathsf{HLS}_n(Y,\bm{X})$, a unified multivariate generating framework built from semistandard Young tableaux and Hall--Littlewood polynomials, to solve a spectrum of lattice- and representation-theoretic counting problems. By proving a broad self-reciprocity formula and establishing that key specialization families (affine Schubert series, Hecke series, Hermite--Smith series, and quiver representation zeta functions) arise from $\mathsf{HLS}_n$ via explicit substitutions, the paper forges deep links between combinatorics, $p$-adic integration, and geometric topology. It further connects these series to affine Bruhat–Tits structures, Bruhat orders, and Stanley--Reisner rings, including a Cohen–Macaulay/topological ball result for the tableau poset $\mathsf{T}_n$, and provides concrete new formulae for Hecke and quiver zeta functions. The analysis leverages a fiber-enumeration approach using intersection and projection tableaux, Dyck-word statistics, and p-adic methods to reveal uniformity in residue-field dependence and to illuminate intricate dualities between seemingly disparate counting problems. Overall, the work greatly broadens the applicability of Hall--Littlewood combinatorics to $p$-adic and representation-theoretic enumerations, yielding new analytic, algebraic, and topological insights.

Abstract

We introduce multivariate rational generating series called Hall-Littlewood-Schubert ($\mathsf{HLS}_n$) series. They are defined in terms of polynomials related to Hall-Littlewood polynomials and semistandard Young tableaux. We show that $\mathsf{HLS}_n$ series provide solutions to a range of enumeration problems upon judicious substitutions of their variables. These include the problem to enumerate sublattices of a $p$-adic lattice according to the elementary divisor types of their intersections with the members of a complete flag of reference in the ambient lattice. This is an affine analog of the stratification of Grassmannians by Schubert varieties. Other substitutions of $\mathsf{HLS}_n$ series yield new formulae for Hecke series and $p$-adic integrals associated with symplectic $p$-adic groups, and combinatorially defined quiver representation zeta functions. $\mathsf{HLS}_n$ series are $q$-analogs of Hilbert series of Stanley-Reisner rings associated with posets arising from parabolic quotients of Coxeter groups of type $\mathsf{B}$ with the Bruhat order. Special values of coarsened $\mathsf{HLS}_n$ series yield analogs of the classical Littlewood identity for the generating functions of Schur polynomials.

Figures (9)

• Figure 1.1: The dual star quiver $\mathsf{S}_4^*$
• Figure 3.1: Valuation and gap partitions for $(\lambda, \mu)\in\mathop{\mathrm{HoSt}}\nolimits_6$
• Figure 3.2: An illustration of the jigsaw operation \ref{['eqn:dual-partition']}
• Figure 3.3: Data associated with an example $(\lambda,\mu)\in\mathop{\mathrm{HoSt}}\nolimits_9$.
• Figure 4.1: An illustration of the jigsaw operation on tableaux

Theorems & Definitions (119)

• Definition 1.1
• Definition 1.2
• Remark 1.3
• Example 1.4: $\mathsf{HLS}_n$ for $n\leqslant 3$
• Theorem A
• Definition 1.5
• Remark 1.6
• Theorem B
• Theorem C
• Corollary 1.7